it appears to be a panelled door.
the way they work is that each of the pieces of wood it is made of will expand or contract individually (shrinking in summer when it dries out).
When it was new, each of the panels was fitted into grooves or slots in the timbers around it, and could move slightly within them. After fifty or so coats of paint, the paint gets into the joints and glues them together so they can no longer move and accommodate movement, so the wood cracks instead.
If you look closely at the edges of that panel, you will find there is a build-up of old paint. You may see that the paint has cracked in places where it has been unable to resist the movement of the wood. The trick is to break the bond this paint makes. A small, sharp triangular scraper, pulled down the joint, will crack away the paint and when it is gone you will be able to see the frame of stiles and rails, and the way panel fits into grooves. Timber expands and contracts across its width, not its length, so when it is restrained, these cracks will be parallel with the grain. The panels in your door are quite large, so they might be made from two (or more) pieces of wood jointed together, and the crack might be where one of these joints has opened up.
Once the paint preventing movement has been removed, you may be able to glue or fill the gap and paint over it. Only the thinnest amount of paint should be applied to the joints round the panel, to prevent it being glued rigid again. This will show less if the paint is somewhat similar to the colour of the wood, or if the door is stained.
In short, any rigid filler will crack again. If you use something elastic like a silicone rubber sealant, very thinly, it may be able to absorb the movement. If the crack is full of rigid filler or hard paint, it will push the two halves of the panel further apart in winter, when it becomes more humid and tries to close up, and the crack will be bigger the following summer. Painting in dark colours will make the cracks less obvious, but a wooden door exposed to summer sun should not be painted black, because it will absorb the sun's heat and get hotter, and drier, and will crack more.
Here endeth….